{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# p25: Stability regions for ODE formulas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format='svg'\n",
    "from numpy import pi,real,imag,zeros,exp,arange\n",
    "from matplotlib.pyplot import figure,subplot,plot,axis,grid,title"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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   "source": [
    "figure(figsize=(8,8))\n",
    "\n",
    "# Adams-Bashforth\n",
    "subplot(2,2,1)\n",
    "z = exp(1j*pi*arange(0,201)/100); r = z - 1\n",
    "s = 1; rr = r/s; plot(real(rr),imag(rr))\n",
    "s = (3 - 1/z)/2; rr = r/s; plot(real(rr),imag(rr))\n",
    "s = (23 - 16/z + 5/z**2)/12; rr = r/s; plot(real(rr),imag(rr))\n",
    "axis('equal'); grid('on')\n",
    "title('Adams-Bashforth')\n",
    "\n",
    "# Adams-Moulton\n",
    "subplot(2,2,2)\n",
    "s = (5*z + 8 - 1/z)/12; rr = r/s; plot(real(rr),imag(rr))\n",
    "s = (9*z + 19 - 5/z + 1/z**2)/24; rr = r/s; plot(real(rr),imag(rr))\n",
    "s = (251*z + 646 - 264/z + 106/z**2 - 19/z**3)/720; rr = r/s; plot(real(rr),imag(rr))\n",
    "d = 1 - 1/z\n",
    "s = 1 - d/2 - d**2/12 - d**3/24 - 19*d**4/720 - 3*d**5/160; dd = d/s; plot(real(dd),imag(dd))\n",
    "axis('equal'); grid('on')\n",
    "title('Adams-Moulton')\n",
    "\n",
    "# Backward differentiation\n",
    "subplot(2,2,3)\n",
    "r = 0\n",
    "for i in range(1,7):\n",
    "    r = r + d**i/i; plot(real(r),imag(r))\n",
    "axis('equal'); grid('on')\n",
    "title('Backward differentiation')\n",
    "\n",
    "# Runge-kutta\n",
    "subplot(2,2,4)\n",
    "w = 0; W = 1j*zeros(len(z)); W[0] = w;\n",
    "for i in range(1,len(z)):\n",
    "    w = w - (1+w-z[i]); W[i] = w\n",
    "plot(real(W),imag(W))\n",
    "w = 0; W = 1j*zeros(len(z)); W[0] = w;\n",
    "for i in range(1,len(z)):\n",
    "    w = w - (1+w+0.5*w**2-z[i]**2)/(1+w); W[i] = w\n",
    "plot(real(W),imag(W))\n",
    "w = 0; W = 1j*zeros(len(z)); W[0] = w;\n",
    "for i in range(1,len(z)):\n",
    "    w = w - (1+w+0.5*w**2+w**3/6-z[i]**3)/(1+w+0.5*w**2); W[i] = w\n",
    "plot(real(W),imag(W))\n",
    "w = 0; W = 1j*zeros(len(z)); W[0] = w;\n",
    "for i in range(1,len(z)):\n",
    "    w = w - (1+w+0.5*w**2+w**3/6+w**4/24-z[i]**4)/(1+w+w**2/2+w**3/6); W[i] = w\n",
    "plot(real(W),imag(W))\n",
    "axis('equal'); grid('on')\n",
    "title('Runge-Kutta');"
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